Sequence of dense sets in Banach spaces

Question

Let $A_0 \supset A_1 \supset A_2 \supset \cdots$ - sequence of embedded Banach spaces and $B_0 \supset B_1 \supset B_2 \supset \cdots$ - suquence of linear spaces such that $B_i$ dense in $A_i$, it is true that $\bigcap_{k=0}^\infty B_k = 0 \Rightarrow \bigcap_{k=0}^\infty A_k = 0$ ?

Answer 1

No. Let $\{q_m\}$ be an enumeration of rationals in $[0,1]$, and define $B_n$ to be the space of continuous functions that vanish at $q_1,\dots,q_n$. Then $B_1\supset B_2\supset \dots$ and $\bigcap_n B_n = \{0\}$.

On the other hand, each $B_n$ is dense in $L^2([0,1])$, so one can take $A_n=L^2([0,1])$ for every $n$.